Optimal. Leaf size=51 \[ \frac {c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{5/2}}+\frac {c}{2 a^2 x^2}-\frac {1}{6 a x^6} \]
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Rubi [A] time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 325, 205} \[ \frac {c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{5/2}}+\frac {c}{2 a^2 x^2}-\frac {1}{6 a x^6} \]
Antiderivative was successfully verified.
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Rule 205
Rule 275
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^7 \left (a+c x^4\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+c x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {1}{6 a x^6}-\frac {c \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+c x^2\right )} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {1}{6 a x^6}+\frac {c}{2 a^2 x^2}+\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 a^2}\\ &=-\frac {1}{6 a x^6}+\frac {c}{2 a^2 x^2}+\frac {c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 88, normalized size = 1.73 \[ -\frac {3 c^{3/2} x^6 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+3 c^{3/2} x^6 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+\sqrt {a} \left (a-3 c x^4\right )}{6 a^{5/2} x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 112, normalized size = 2.20 \[ \left [\frac {3 \, c x^{6} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{4} + 2 \, a x^{2} \sqrt {-\frac {c}{a}} - a}{c x^{4} + a}\right ) + 6 \, c x^{4} - 2 \, a}{12 \, a^{2} x^{6}}, -\frac {3 \, c x^{6} \sqrt {\frac {c}{a}} \arctan \left (\frac {a \sqrt {\frac {c}{a}}}{c x^{2}}\right ) - 3 \, c x^{4} + a}{6 \, a^{2} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 43, normalized size = 0.84 \[ \frac {c^{2} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a^{2}} + \frac {3 \, c x^{4} - a}{6 \, a^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 43, normalized size = 0.84 \[ \frac {c^{2} \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, a^{2}}+\frac {c}{2 a^{2} x^{2}}-\frac {1}{6 a \,x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.09, size = 43, normalized size = 0.84 \[ \frac {c^{2} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a^{2}} + \frac {3 \, c x^{4} - a}{6 \, a^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 40, normalized size = 0.78 \[ \frac {c^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{2\,a^{5/2}}-\frac {\frac {1}{6\,a}-\frac {c\,x^4}{2\,a^2}}{x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.53, size = 90, normalized size = 1.76 \[ - \frac {\sqrt {- \frac {c^{3}}{a^{5}}} \log {\left (- \frac {a^{3} \sqrt {- \frac {c^{3}}{a^{5}}}}{c^{2}} + x^{2} \right )}}{4} + \frac {\sqrt {- \frac {c^{3}}{a^{5}}} \log {\left (\frac {a^{3} \sqrt {- \frac {c^{3}}{a^{5}}}}{c^{2}} + x^{2} \right )}}{4} + \frac {- a + 3 c x^{4}}{6 a^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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